Paul Dacre, the editor of the Daily Mail, and Telegraph Media Group and News International executives, are concerned that a candidate such as Hacked Off's Brian Cathcart might apply to be on the board of the new press watchdog and the editors insist they need a veto on appointments to stop anyone with an agenda getting a seat at the top table.

Cathcart, who is executive director of Hacked Off, the group campaigning for tighter press regulation, claimed on Thursday that Cameron had acquiesced to demands by the three newspaper groups that they should have the final say over who was on the new regulatory body's board.

But sources in the Mail, Telegraph and News International camp said this is a legitimate concern to stop the likes of Cathcart or "some sub-editor who has borne a grudge against their employer" getting on the board and using their position "to pursue their own narrow agenda rather than regulate and adjudicate on the merits of the cases that come before it".

Under the Leveson recommendations, no serving editor would sit on the board of the press watchdog that will replace the Press Complaints Commission in order to ensure its independence from the industry. However former editors can apply and Cathcart would be eligible as a former deputy editor of the Independent on Sunday.

"The process is open and anyone can apply, but the concern was you might end up with a situation with for instance Brian Cathcart applied as one of the industry members who can't be serving editors but you can't have anyone too junior either. The industry people need to be people who work in the industry and know and understand it," said one newspaper industry source.

Others who it is believed might apply for the job include the former News of the World executive editor Neil Wallis, who might have the support of the tabloids but not the broadsheets, the source said.

The newspaper industry insiders said that the veto they want on appointments is borne out of real concerns and not a desire to return to the ways of the past when the PCC was seen as a poodle of the tabloids and failed to properly investigate allegations about News of the World phone hacking raised by the Guardian in 2009.

They have asked prime minister David Cameron to approve a system of a "qualified majority" whereby all appointments to the new press regulator's board must have majority approval of the seven independent members making the appointments decisions and a majority of the five industry members.

The closed-door talks between Cameron and individual editors including Dacre has also led to a split in the press industry, which had initially agreed to set aside their differences over a breakfast summit at the Delaunay restaurant in London in December and implement most of the Leveson report recommendations, including the creation of a new press regulatory body with the powers to exact fines of up to £1m.

"We don't like it, but we are going to have to implement it," one tabloid national newspaper editor remarked to the Guardian at the time.

A little over three months later the Delaunay pact appears to be in tatters with old rivalries resurfacing between newspapers and Westminster now locked in a do-or-die battle over the new press regulator.

According to one newspaper insider involved in the talks with Downing St, it all started to unravel on 5 February, the day Lord Puttnam won Lords backing to tack some Leveson amendments to the defamation bill.

At that point the political momentum had dissipated around Leveson as the Tories delayed on their proposals for a new watchdog, prompting Puttman's manoeuvre. But behind the scenes, newspaper groups, notoriously divided over the PCC's past performance, had managed to agree on almost everything, something they considered a major feat.

"We were within an inch of agreeing the recognition criteria [for the new press regulator] and the only outstanding issue was one sentence on group complaints," said one newspaper source involved in the talks.

"Cameron had asked us to find agreement when we went to see him in number 10 after Leveson, he did his spiel and left us with [culture secretary] Maria Miller and [PM's policy adviser] Oliver Letwin. She asked us all to go away and find agreement which we did," said the source.

"It is clear to us that closed-door negotiations with the Conservatives have so far failed to generate a politically acceptable outcome and the process has alienated stakeholders in the debate, including party leaders and parliamentarians," said the editors of the respective papers, Alan Rusbridger, Lionel Barber and Chris Blackhurst.

Their letter was sent to former Mail on Sunday editor Peter Wright and Paul Vickers, company secretary of Trinity Mirror, who have been leading negotiations on behalf of the industry.